Toward a Stronger Constraint for Non-Trivial Inconsistent Theories

Antônio Mesquita

doi:10.12775/LLP.2025.015

Authors

Antônio Mesquita Department of Logic and Philosophy of Science, Autonomous University of Madrid https://orcid.org/0000-0002-7649-599X

DOI:

https://doi.org/10.12775/LLP.2025.015

Keywords

ECQ, explosion, inconsistency, triviality, paraconsistency

Abstract

This article discusses the definition of paraconsistency understood as the property of a consequence relation that does not trivialize inconsistent theories. Some logicians have argued that standard paraconsistency, the requirement of a non-explosive consequence relation, is insufficient for that purpose. In this article, we have a twofold goal. First, we offer an exposition of some attempts to strengthen standard paraconsistency in the literature. After discussing the shortcomings of those attempts, we examine the concepts of triviality in relation to which those concepts of paraconsistency are defined. Then, as our second goal, we propose an alternative definition of paraconsistency that aims to avoid the trivialization of inconsistent theories in a stricter sense

References

Barrio, E., F. Pailos, and D. Szmuc, 2018, “What is a paraconsistent logic?”, pages 89–108 in W. Carnielli and J. Malinowski (eds.), Contradictions, from Consistency to Inconsistency, Trends in Logic, vol. 47, Springer. DOI: https://doi.org/10.1007/978-3-319-98797-2_5

Batens, D., 1980, “Paraconsistent extensional propositional logics”, Logique et Analyse 90/91: 195–234.

Béziau, J.-Y., 2000, “What is paraconsistent logic?”, pages 95–111 in D. Batens, C. Mortensen, G. Priest and J.-P. Van Bendegem (eds.), Frontiers of Paraconsistent Logic, Research Studies Press.

Carnielli, W., M. E. Coniglio, and J. Marcos, 2007, “Logics of formal inconsistency”, pages 1–93 in D. M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. 14, Springer. DOI: https://doi.org/10.1007/978-1-4020-6324-4_1

Castiglioni, J. L., and R. E. B. Biraben, 2013, “Strict paraconsistency of truthdegree preserving intuitionistic logic with dual negation”, Logic Journal of the IGPL 22 (2): 268–273. DOI: https://doi.org/10.1093/jigpal/jzt027

Da Costa, N. C. A., 1974, “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic 15 (4): 497–510. DOI: https://doi.org/10.1305/ndjfl/1093891487

Da Ré, B., M. Rubin, and P. Teijeiro, 2022, “Metainferential paraconsistency”, Logic and Logical Philosophy 31 (2): 235—260. DOI: https://doi.org/10.12775/LLP.2022.008

Michael, M., 2016, “On a “most telling” argument for paraconsistent logic”, Synthese 193 (10): 3347–3362. DOI: https://doi.org/10.1007/s11229-015-0935-6

Pailos, F., and B. Da Ré, 2023, Metainferential Logics, Trends in Logic, vol. 61, Springer. DOI: https://doi.org/10.1007/978-3-031-44381-7

Priest, G., 1979, “The logic of paradox”, Journal of Philosophical Logic 8 (1): 219–241. DOI: https://doi.org/10.1007/BF00258428

Priest, G., 2002, “Paraconsistent logic”, pages 287–393 in D. M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. 6, Springer. DOI: https://doi.org/10.1007/978-94-017-0460-1_4

Priest, G., 2007, “Paraconsistency and dialetheism”, pages 129–204 in D. M. Gabbay and J. Woods (eds.), Handbook of the History of Logic, vol. 8, North-Holland. DOI: https://doi.org/10.1016/S1874-5857(07)80006-9

Priest, G., and R. Routley, 1983, On Paraconsistency, Research Series in Logic and Metaphysics, Australian National University.

Priest, G., K. Tanaka, and Z. Weber, 2022, “Paraconsistent logic”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford University. http://plato.stanford.edu/archives/spr2022/entries/logic-paraconsistent/

Pulcini, G., and A. C. Varzi, 2018, “Paraconsistency in classical logic”, Synthese 195: 5485–5496. DOI: https://doi.org/10.1007/s11229-017-1458-0

Ripley, D., 2015, “Paraconsistent logic”, Journal of Philosophical Logic 44 (6): 771–780. DOI: https://doi.org/10.1007/s10992-015-9358-6

Robles, G., 2009, “Weak consistency and strong paraconsistency”, tripleC 7 (2): 185–193. DOI: https://doi.org/10.31269/triplec.v7i2.99

Robles, G., and J. Méndez, 2009, “Strong paraconsistency and the basic constructive logic for an even weaker sense of consistency”, Journal of Logic, Language, and Information 18 (3): 357–402. DOI: https://doi.org/10.1007/s10849-009-9085-x

Urbas, I., 1990, “Paraconsistency”, Studies in Soviet Thought 39 (3/4): 343–354. DOI: https://doi.org/10.1007/BF00838045